Method of and apparatus for correction of laser beam phase front aberrations

ABSTRACT

In accordance with one embodiment of the present invention, a controlling laser beam at a first wavelength is introduced into an interferometer such that one half of it contains the aberrations of the laser beam to be controlled and the other half is and remains undistorted. The aforementioned two halves of the controlling laser beam are combined at the output of the interferometer and then, together with the laser beam to be controlled at a second wavelength is introduced collinearly into a gas cell or phase corrector. Upon combination or interference of the two halves of the controlling laser beam the phase-front aberrations are translated to intensity modulations across the cross section of the beam. Utilizing nonlinear dispersion in the gas cell in accordance with the invention, the intensity modulations across the controlling laser beam are transformed by nonlinear dispersion interaction into appropriate refractive index variations at the second wavelength which in turn causes corresponding phase changes across the phase front of the controlled laser beam. Adjustment of path length in the interferometer permits compensation of the aberrations in the controlled laser beam.

This invention relates to control of laser beams and more particularly to the correction of laser beam phase front aberrations.

DISCUSSION OF PRIOR ART

It is known that the introduction of a light beam at an appropriate first wavelength in a suitably selected gaseous medium under collision free conditions will induce a change in the refractive index of the medium for light at a second wavelength, where the change in the refractive index is proportional to the local intensity of the light beam at the first wavelength. This phenomenon is known as and referred to as nonlinear dispersion.

It has been observed experimentally as reported by D. Grischkowsky, "Coherent Excitation, Incoherent Excitation, and Adiabatic States," Phys. Rev. A14, 801-812 (1976) and J. E. Bjorkholm and P. F. Liao, "Line Shape and Strength of two-photon Absorption in Atomic Vapor with a Resonant or Nearly Resonant Intermediate State," Phys. Rev. A14 751-760 (1976) that nonlinear dispersion exists in a gaseous medium, i.e., the effect of the radiation field at one frequency on the refractive index of a gaseous medium at a different frequency. See also "Probing Adiabatic States," A. Flusberg and S. R. Hartman, Phys. Rev. A14, 813-815 (1976) which supports the analysis of Grischkowsky. The experiments were carried out and the observations thereof were explained by a perturbation analysis, using the adiabatic approximation and a collision-free condition in the gaseous medium.

Heretofore, nonlinear dispersion has not been of any particular value or considered useful for practically operative laser controlled optics because, inter alia, under the collision-free conditions with which it has been used substantially complete deflection of an intense laser beam necessitates an intense control laser beam with at least substantially the same photon number density as that of the beam to be deflected.

Numerous inexpensive techniques other than nonlinear dispersion are available for deflection of low intensity laser beams and nonlinear dispersion was not considered feasible or desirable for reasons including that noted immediately hereinabove.

SUMMARY OF THE INVENTION

We have found, however, that under collision-dominated conditions in accordance with one feature of the invention, the power requirement of the field generating dispersion is substantially smaller than the power of the beam to be deflected and that it is possible to at least substantially, if not totally, deflect an appropriate laser beam or beams including high power laser beams without having to invest an unreasonably large amount of power in the controlling laser beam as was required heretofore. This is because, broadly, we have found that under collision dominated conditions the effect of unit intensity of the controlling laser beam at its frequency on the refractive index at the controlled laser beam frequency is much larger that the effect of unit intensity of the controlled laser beam on the refractive index at the frequency of the controlling laser beam.

In accordance with the present invention real-time laser-controlled refractive optical techniques and device may be provided using nonlinear dispersion in a gaseous medium. Refractive optical elements may be provided that function as spherical or curved reflectors and refractive lenses or prisms for expanding or contracting a laser beam or beams, as well as provide gratings that function to provide partial or total reflection or deflection of a laser beam. The generation of a grating will permit at least substantial, if not total, deflection of large cross section and powerful laser beams by the investment of a moderate amount of power. Since the response time of the gaseous medium is dependent on the lifetimes involved and on gaseous diffusion which can be made as short as a nanosecond, fast acting devices for high power lasers can also be provided. Real-time color holographic displays are also possible.

In accordance with the present invention, real-time laser-controlled refractive optical techniques and devices may be provided that, inter alia, permit real-time correction of rapid phase front distortions or aberrations of high power laser beams.

For a discussion of a prior art technique and apparatus for laser beam wave front phase compensation and shaping, reference is made to "Technique for Real-Time High Resolution Adaptive Phase Compensations," A. D. Fisher and C. Warde, Optics Letters, Volume 8, Number 7, 353-355, July 1983. This phase-measurement and compensation technique utilizes an interference loop in conjunction with a monolithic optically addressed spatial light modulator. The spatial light modulator may typically comprise an electro-optic crystal, dielectric mirror, microchannel plate and a photocathode.

In accordance with one embodiment of the present invention, a controlling laser beam at a first wavelength is introduced into an interferometer such that one half of it contains the aberrations of the laser beam to be controlled and the other half is and remains undistorted. These two halves of the controlling laser beam are combined at the output of the interferometer and then, together with the laser beam to be controlled at a second wavelength is introduced collinearly into a gas cell or phase corrector. Upon combination or interference of the two halves of the controlling laser beam, the phase-front aberrations are translated to intensity modulations across the cross section of the beam. Utilizing nonlinear dispersion in the gas cell in accordance with the invention, the intensity modulations across the controlling laser beam are transformed by nonlinear dispersion interaction into appropriate refractive index variations at the second wavelength which in turn causes corresponding phase changes across the phase front of the controlled laser beam. Adjustment of path length in the interferometer permits compensation of the aberrations in the controlled laser beam.

BRIEF DESCRIPTION OF THE DRAWINGS

The novel features that are considered characteristic of the invention are set forth in the appended claims; the invention itself, however, both as to its organization and method of operation, together with additional objects and advantages thereof, will best be understood from the following description of a specific embodiment when read in conjunction with the accompanying drawings, in which:

FIG. 1 is a diagram showing energy levels of a gas suitable for use in accordance with the invention;

FIG. 2 is a diagrammatic representation of a gaseous medium functioning as a grating;

FIG. 3 is a diagrammatic representation of a gaseous medium functioning as a grating having grating planes at a slant angle φ;

FIG. 4 is a diagrammatic representation of a gaseous medium functioning as a lens;

FIG. 5 is an illustration of one form of gas cell for a gaseous medium in accordance with the invention; and

FIG. 6 is a schematic diagram of apparatus for correcting aberrations imposed on a laser beam.

DETAILED DESCRIPTION OF THE INVENTION

To facilitate understanding of the invention, the most relevant principles of nonlinear dispersion including nonlinear dispersion under collision dominated conditions will now be discussed. By way of example an atomic or molecular gaseous medium suitable for use in accordance with the present invention (more fully discussed hereinafter) may have energy levels as shown in FIG. 1 where the 0→1 and 1→2 transitions are allowed electric dipole transitions. If such a suitable gas of atoms is irradiated by two radiation fields ε₁ (ω₁) and ε₂ (ω₂), where ω₁ is off resonance from the 0→1 transition by δ_(o), ω₂ is off resonance from the 1→2 transition by Δ_(o) and the sum of the frequencies is off resonance from the 0→2 two-photon transition by Ω_(o) =δ_(o) +Δ_(o), the interaction with the two fields will induce a nonlinear refractive index of the medium for both frequencies, where the refractive index Δn^(NL) (ω₂) is proportional to the intensity at ω₂ and the refractive index at Δn^(NL) (ω₁) is proportional to the intensity at ω₁.

The nonlinear polarization induced in the medium, may be calculated starting with the equation for the density matrix ρ of the medium: ##EQU1## where H=H_(o) +V, H_(o) is the Hamiltonian of the free atoms ρ_(kg) is the kj matrix element of ρ, i is equal to (-1)^(1/2), h is the ratio of Planck's constant to 2π, τ_(kj) is the decay time of ρ_(kj), V is the interaction Hamiltonian, and k and j are indices taking on the values 0, 1 or 2. The last term in Equation (1) is introduced phenomenologically to account for the homogeneous relaxation of the excited states.

The interaction Hamiltonian V is given by:

    V=-(μ·ε)=-μ.sup.x ·ε

where μ is the electric dipole moment operator, x is the polarization direction of the field and

    ε=ε.sub.1 +ε.sub.2 =1/2(E.sub.1 e.sup.iω.sbsp.1.sup.t +E.sub.2 e.sup.iω.sbsp.2.sup.t)+c.c.

where ε is the total electric field of the light, ε₁ is the electric field at ω₁, ε₂ is the electric field at ω₂, E₁ and E₂ are slowly varying amplitudes, t represents time, and c.c means complex conjugate.

Insertion of the Hamiltonian in Equation (1) gives four equations for the density matrix elements: ##EQU2## where hω_(nm) is the energy difference between levels n and m; and n,m are indices which may take on the values 0, 1 and 2. Equations (2a)-(2d) may be solved with the following assumptions:

(a) None of the transitions are saturated, i.e.,

    ρ.sub.11 <<ρ.sub.00 and ρ.sub.22 <<ρ.sub.11 ;

(b) ρ₁₀ interacts more strongly with ε₁ than with ε₂ ; and

(c) the interaction 1→2 is weaker than the 0→1, i.e.,

    |ρ.sub.12 V.sub.21 -ρ.sub.21 V.sub.12 |<<|ρ.sub.10 V.sub.01 -V.sub.10 ρ.sub.01 |.

Inserting the fields in Equations (2a)-(2d), retaining only near resonance terms, substituting ##EQU3## and making the slowly-varying-amplitude approximation: ##EQU4## Equations (2a)-(2d) may be reduced to four algebraic equations for the amplitudes of the density matrix elements.

The polarization of the gaseous medium is given by ##EQU5## where the symbol Σ denotes a sum.

If the solutions of the algebraic equations are substituted for the density matrix elements, and the real and imaginary parts of the susceptibility are separated, we obtain that the nonlinear refractive index Δn^(NL) (ω₂) at frequency ω₂ is given by the formula: ##EQU6## where μ₀₁ ^(X) is the 0-1 matrix element of the x component of the dipole operator, ##EQU7## and τ_(col) is defined by: ##EQU8## From Equations (2a)-(2d) it will be clear that τ₁₁ measures the decay rate of the population in the first excited state, while π₁₀ measures the decay rate of the phase of this state. In the absence of collisions it may be shown that ##EQU9## it having been assumed in Equation (5) that the collisions affects the phase of the first excited state rather than its population density.

Equations (4) and (5) may now be used to differentiate between two different regimes: (a) the prior art collision-free regime in which τ_(col) >>2τ₁₁ and (b) the collision-dominated regime of the present invention where τ_(col) <<τ₁₁.

From Equation (4) it can be shown that under collision-free conditions:

    sign (Δn)=sign (Ω.sub.o)                       (6)

Stated differently, Equation (6) says that normal dispersion behavior exists near a resonance transition with the resonance corresponding to a true simultaneous two-photon transition. In this case, the proximity of ω₁ to the 0→1 resonance affects the magnitude of Δn^(NL) (ω₂), but does not effect the sign of Δn^(NL) (ω₂), which is only dependent on the sign of 106 _(o).

On the other hand in the collision dominated regime (τ_(col) <<τ₁₁) of the present invention, it can be shown from Equation (4) that:

    sign (Δn)=sign (Δ.sub.0)                       (7)

This corresponds to a sequence of two single-photon transitions, 0→1 followed by 1→2, disturbed on the way by dephasing collisions. In fact it may be shown that the nonlinear refractive index in this case is proportional to the actual population density in level 1.

If now Equation (4) is expressed in terms of the absorption coefficient γ(ω₁) for the beam at frequency ω₁ : ##EQU10## where λ₁ =2πc/ω₁ is the wavelength corresponding to ω₁. Similarly, calculating the nonlinear refractive index at frequency ω₁ gives: ##EQU11## or in short Equations (8) and (9) can be written:

    Δn.sup.NL (ω.sub.2)=β.sub.1 |E.sub.1 |.sup.2                                          (10)

    Δn.sup.NL (ω.sub.1)=β.sub.2 |E.sub.2 |.sup.2                                          (11)

It may be noticed from Equations (10) and (11) that under collision-free conditions [(2τ₁₁ Δ_(o) /τ_(col) |Δ|²)<<|Ω_(o) |/|Ω|² ] and off resonance (|δ_(o) |>>1/τ₁₀ ; |Ω_(o) |>>1/τ₂₀), that the coefficients β₁ and β₂ are equal, whereas under collision-dominated conditions ((2τ₁₁ |Δ_(o) |/τ_(col) |Δ|²)>>|Ω_(o) |/|Ω|²) we have |β₁ |>>|β₂ |. The implication of the above noted differences is the basis for selecting collision-dominated conditions for laser-controlled optics applications in accordance with the invention.

As previously noted, nonlinear dispersion in gaseous media may be used in accordance with the invention to control the phase front of a laser beam at wavelength λ₂, by another laser beam at wavelength λ₁. Real-time laser-controlled refractive optical elements as well as real-time laser-controlled holographic optical elements can also be provided.

A real-time laser-controlled refractive lens or prism can be generated by appropriate tailoring of the spatial intensity distribution of the controlling laser beams in the gaseous medium. From the preceding discussion it will be clear that only the collision-dominated regime is suitable for the types of application here concerned. This is because in the collision-dominated regime |β₁ |>>|β₂ | and therefore the effect of unit intensity of a controlling laser beams at ω₁ on the refractive index at ω₂ is much larger than the effect of unit intensity of a controlled laser beam on the refractive index at the controlling laser beam at (ω₁) as shown by Equations (10) and (11).

An analysis of power requirement for refractive optics shows that the power requirement of the laser for providing the controlling laser beam in accordance with the present invention is strongly dependent on the diameter of the laser beam to be controlled. This is because in an ordinary refractive lens, the phase lag in the center of the beam, compared to the beam boundary is a quadratic function of the beam diameter.

Consideration will now be given to the application of the present invention to real-time laser-controlled holographic optics.

Real-time holographic optical elements for a laser beam at frequency ω₂ may be provided, using nonlinear dispersion in an appropriate gaseous medium in accordance with the invention, if the holographic image of the desired optical element is first produced in a conventional manner in the medium, by interfering two coherent laser beams of frequency ω₁ having suitable phase fronts. The spatial intensity modulation of the field at ω₁ produced by the interference will cause an identical spatial modulation of the refractive index of the gaseous medium at the frequency ω₂. Thus, for example, a volumetric planar grating for light of frequency ω₂ can be generated by interfering two non-colinear plane waves of frequency ω₁ in the medium, while a real-time holographic lens can be produced by interfering in the medium a plane wave of frequency ω₁ with a mutually coherent spherical wave of the same frequency.

The response time of such an optical element is determined by the lifetime involved and can be as short as 10 nsec. In fact, spatial diffusion decreases the effective lifetime, thus decreasing response time even further.

Directing attention now to the provision of a real-time holographic grating: Consider as shown in FIG. 2 two coherent laser beams at frequency ω₁, and with equal intensity, impinging at an angle to each other, on a gaseous atomic or molecular medium, with appropriate energy levels and transitions as shown and described in connection with FIG. 1. In FIG. 2 (and FIG. 3), the X=0 plane is the plane where the grating planes start and interaction begins and the plane X=L is the plane where the grating planes end and the interaction stops. If the phase fronts of the fields ε₁ and ε'₁ are reasonably planar in the interaction volume, then the interference between these two fields will generate a real-time holographic grating with sinusoidal spatial modulation of the refractive index at frequency ω₂. Defining ε₁ and ε₁ ' as: ##EQU12## where E₁, E'₁ are complex, underlined variables denote vector quantities; K₁ and K₁ ' are the wave vectors of the two fields at ω₁ and denotes a spatial position; using Equations (8) and (10), Δn(ω₂) can be written as:

    Δn(ω.sub.2)=β.sub.1 [E.sub.1 E'.sub.1 *e.sup.i(K.sbsp.1.sup.-K'.sbsp.1.sup.)r +c.c.]            (12)

The spacing d between two adjacent maxima of the modulation is given by ##EQU13## where α is half the angle between K₁ and K'₁ and n₀ (λ) is the refractive index of the unperturbed gaseous medium at wavelength λ₁.

If a third beam of light at frequency ω₂,[ε₂ (ω₂)] is incident on the volumetric grating at an angle θ to the planes of constant refractive index such that θ satisfies the Bragg condition:

    2n.sub.0 (λ.sub.2)d sin θ=λ.sub.2      (14)

then this third beam will be efficiently deflected by the grating if the gaseous medium is under collision-dominated conditions, as is shown below.

With the formation of ε'₂ (ω₂) by deflection of ε₂ (ω₂) (see FIG. 2) by the real-time grating: the interference between ε₂ (ω₂) and ε'₂ (ω₂) generates a real-time holographic grating with sinusoidal spatial modulation of the refractive index at frequency ω₁, given by:

    Δn(ω.sub.1)=β.sub.2 E.sub.2 E.sub.2 '*{e.sup.i(K.sbsp.2.sup.-K.sbsp.2.sup.')r +c.c.}          (15)

where K₂ and K₂ ' are the respective wave vectors of ε₂ and ε'₂ and β₂ is given by Equations (9) and (11).

The refractive index grating Δn(ω₁) will have the same geometry as that of the refractive index grating Δn(ω₂) generated by the interference of ε₁ (ω₁) and ε'₁ (ω₁). Therefore the grating generated by the fields ε₁ (ω₁) and ε'₁ (ω₁) reflects ε₂ (ω₂) into ε'₂ (ω₂) (or vice versa), while the grating generated by the fields ε₂ (ω₂) and ε'₂ (ω₂) reflects ε₁ (ω₁) into ε'₁ (ω₁) (or vice versa).

Under collision-dominated conditions ε₂ (ω₂) may be almost totally deflected into ε'₂ (ω₂) (unit conversion efficiency) without having to invest an unreasonably larger power at frequency ω₁ as will now be shown.

The combination of Equations (13) and (14) provides: ##EQU14## i.e., the deflection angle (2θ) of the controlled beam can be selected for given λ₁ and λ₂ by choosing an appropriate angle α.

In a real-time grating as described above, there are four fields interacting with each other via the nonlinear polarization of the gaseous medium.

The field equation is: ##EQU15## where ∇ is the gradient operator with components ∂/∂x, ∂/∂y, and ∂/∂z along x, y and z, respectively.

    ε(ω.sub.1)=ε.sub.1 (ω.sub.1)+ε.sub.1 '(ω.sub.1),

    ε(ω.sub.2)=ε.sub.2 (ω.sub.2)+ε.sub.2 '(ω.sub.2)

and

    ε(ω.sub.i)=n.sub.o.sup.2 +2n.sub.o Δn(ω.sub.i).

Writing

    ε.sub.i =1/2[E.sub.i e .sup.i(K.sbsp.i.sup.r-ω.sbsp.i.sup.t) +c.c.],

imposing the Bragg condition:

    K.sub.1 -K.sub.1 '=K=K.sub.2 -K.sub.2 '                    (18)

where K is the "grating wave vector", inserting Δn(ω_(i)) from Equations (12) and (15) into Equation (17), and assuming the slowly varying amplitude approximation (i.e., neglecting terms of the order of (∂² E_(i) /∂x²) provides four coupled equations: ##EQU16##

Using straightforward manipulation, Equations (9a)-(9d) provide a set of equations describing the rate of change of photons in each field: ##EQU17## where

    R=(E.sub.1 *E.sub.1 'E.sub.2 '*E.sub.2)

Im(R) means the imaginary part of R, and ∂/∂r_(i) mean 1/K_(i) (K_(i) ∇), i.e., the gradient along the propagation direction of the appropriate field.

Addition of Equations (20a) to (20b) and Equations (20c) to (20d) provides photon conservation laws: ##EQU18##

Equations (21a) and (21b) state that at any point in the interaction region, photons from the field ε₂ are converted into photons in the field ε'₂ (or vice versa) and photons from the field ε'₁ are converted into photons in the field ε₁ (or vice versa), in such a way that the sum of the photon number densities of ε₂ and ε'₂ as well as the sum of the photon number densities of ε₁ and ε'₁ is conserved.

Subtraction of Equations (20d) from (20a) and Equations (20c) from (20b) provides: ##EQU19##

As has previously been shown, under collision-free conditions β₁ =β₂. As may be seen from Equations (22a) and (22b), under these conditions the rate of change of the photon number density of the field ε₂ equals that of ε'₁. This is to say, under collision-free conditions, for any photon converted from ε₂ into ε'₂, a photon is converted from ε'₁ into ε₁. In this limit the process is precisely a phase-matched four-photon parametric process which is well known in solid and gaseous media. Under these conditions complete deflection of an intense laser beam at frequency ω₂ necessitates an intense laser beam at frequency ω₁ with at least the same photon number density as that of the deflected beam.

The situation is, however, completely different when the gaseous medium is under collision-dominated conditions, i.e., |β₁ |>>|β₂ |. From Equations (21) and (22) it can be seen that under collision dominated conditions an intense field E₂ can be converted into E'₂ with relatively small conversion of E'₁ into E₁. It is emphasized here that under collision-dominated conditions, as will be shown hereinafter, the power requirement of the fields generating the grating (ε₁ and ε'₁), for efficient deflection of the field ε₂, is independent of the power of the deflected field. This contrasts with the prior art four-wave-mixing process (under collision-free conditions) for which the Manley-Row relations dictate that for each reflected photon of ε₂, a photon of ε'₁ has to be converted into ε₁ as discussed above. The above noted deviation from the limitations of a true four-wave-mixing process is possible because under collision-dominated conditions momentum is imparted to the atomic or molecular species. Thus conservation of momentum is not satisfied by the fields themselves and the main contribution to nonlinear dispersion at ω₂ is not by four-wave-mixing, but rather by scattering of photons at frequency ω₂ by atoms (or molecules) in level 1. It should be emphasized here that the above conclusions are true for the collision-dominated regime even if one operates far from resonance on the 0→1 transition, i.e., |δ_(o) |>>1/τ₁₀.

From Equations (22), the condition for large photon conversion of ε₂ into ε'₂ with relatively small conversion of ε'₁ into ε₁ is:

    (β.sub.1 -β.sub.2)Im(R)<0                        (23)

From Equation (20b), the condition for a positive gain for E'₂ is:

    β.sub.1 Im(R)<0                                       (24)

The two requirements of Equations (17) and (18) are compatible under collision-dominated conditions for which

    |β.sub.1 |>>|β.sub.2 |

Under collision-dominated conditions β₁ can be positive or negative depending on the sign of Δ_(o) (see Equation (8)).

Since Im(R)=Im(E₁ *E'₁ E₂ E'₂ *), the sign of Im(R) is dependent on the initial phase of the reflected field E'₂. The inequality of Equation (24) means that for a given β₁ only a field with the appropriate initial phase (relative to the initial phases of E₁, E'₁ and E₂) that satisfies the inequality of Equation (24) will have a positive gain and will be generated.

For a consideration of reflection efficiency, reference is now made to Equation (19) for the fields. Under collision-dominated conditions the attenuation of the fields E'₁ and E₁ is mainly due to absorption losses in the gaseous medium. Assuming that γ(ω₁)L<1, where γ(ω₁) is the absorption coefficient at ω₁ and L is the interaction length, then E₁, E'₁ ˜const and Equations (191)-(19d) reduce to two equations for E₂ and E'₂ : ##EQU20## where Δn_(o) (ω₂) is the amplitude of the sinusoidal refractive index modulation. It is to be noted here that the assumption that the main intensity change in E₁ (and E'₁) is due to absorption is valid only when 2|β₂ |ω₁ |E₂ |² /c<<γ(ω₁) [see Equations (20c) and (20d)]. This puts an upper limit on the intensities |E₂ |² and |E'₂ |² for which Equations (25a) and (25b) are valid. In the following discussion it is assumed that |E₂ |², |E'₂ |² are below the above noted limit. In the more general case of higher intensity the four coupled Equations (19) must be solved.

Equations (25) are the same as those describing a lossless volumetric dielectric grating and these equations have been solved by Kogelnik in the plane-wave approximation for a transmission-grating geometry as well as for a reflection-grating geometry.

Before discussing the solutions, the two geometries, using the slant angle φ as introduced by Kogelnik and shown in FIG. 3 will be defined. As in FIG. 2 the x=0 plane is the plane where the grating planes start and is the beginning of the interaction region and the plane x=L is the plane where the grating planes end and the interaction region ends. φ is the slant angle, which is the angle between the normal to the grating planes and the x axis. θ is the angle between K₂ and the grating planes, and it is assumed that θ satisfies the Bragg condition. γ is the angle between K'₂ (the wave vector of the deflected or reflected beam) and the x-axis.

A transmission grating is provided when γ<π/2 and a reflection grating is provided when γ>π/2. It can be shown from geometrical considerations that one gets a transmission geometry when θ<φ, and a reflection geometry when θ>φ. In particular, when φ=π/2 there exists a transmission geometry for all 0≦θ<π/2, and when φ=0 there exists a reflection geometry for all values of 0<θ≦π/2.

Referring now to Equations (25) and using Kogelnik's solution the deflection efficiency in the transmission geometry is defined by: ##EQU21## where θ is assumed to satisfy the Bragg condition of Equation (16). Similarly, the reflection efficiency for the reflection geometry may be shown to be: ##EQU22##

For transmission geometry θ<φ and for reflection geometry θ>φ, so both expressions are well defined.

If, for the sake of simplicity, it is assumed φ=π/2 for the transmission geometry and φ=0 for the reflection geometry, then: ##EQU23## where θ is the Bragg angle for the field ε₂, and Δn₀ is the amplitude of the sinusoidal refractive index modulation at frequency ω₂, generated by the fields ε'₁ and ε₁.

Consideration will now be given to transmission geometry (beam deflection with φ=π/2) using Equation (28) for deflection efficiency. Modification of the results for reflection geometry (using Equations (29) or (27)) is straightforward.

Under collision-dominated conditions in accordance with the invention the effect of diffusion can be accounted for by writing Equation (28): ##EQU24## Here η_(D) is an efficiency factor determined by diffusion and given by: ##EQU25## where τ_(D) is the characteristic time constant for the dissipation of the real-time grating by diffusion.

The diffusion equation for a refractive index grating is: ##EQU26## where y is the direction of the grating modulation (see FIG. (2)) and D is the diffusion constant in the gaseous medium. Writing Δn as:

    Δn=Δn.sub.o cos (Kr)                           (33)

and noting that n_(o) (λ₁)˜1, so that ##EQU27## we get for the diffusion time constant: ##EQU28## Diffusion can be neglected only if τ_(D) >>τ₁₁, i.e., when: ##EQU29##

Equation (35) set forth immediately hereinabove states that for a given gaseous medium diffusion can be neglected if the angle of incidence α of the beams generating the real time grating is small enough (i.e., the spacing between the grating maxima is large enough). It is to be remembered here that a small α means also a relatively small deflection angle θ by virtue of Equation (16). For larger angles diffusion will cause an increase of the power requirements to compensate for the additional dissipation of the refractive index modulation, as will be shown hereinafter.

From Equation (30) it will be seen that under collision dominated conditions in accordance with the invention unit conversion efficiency of E₂ into E'₂, i.e., total deflection of a beam at frequency ω₂ impinging at the Bragg angle θ is possible if ##EQU30##

Considering now intensity requirements, the intensity I₁ of the field at frequency ω₁, required for total deflection of the beam at ω₂ (see FIG. (2)) can be calculated from Equation (36). From Equation (8) Δn₀ can be written as:

    Δn.sub.0 =Aγ(ω.sub.1)τ.sub.11 I .sub.1 (37)

where ##EQU31## f₁₂ is the oscillator strength for the 1→2 transition; υ_(ij) is the energy separation between the corresponding levels, all energies are measured in cm⁻¹ and I₁ in Equation (37) is in units of W/cm². Substituting Equation (37) into Equation (36) and requiring that ##EQU32## i.e., approximately constant refractive index modulation amplitude in the medium, it may be concluded that the intensity I₁ at frequency ω₁ necessary to totally deflect a beam at frequency ω₂ impinging at the Bragg angle θ has to satisfy ##EQU33## where in deriving of Equation (40) it has been assumed for the sake of simplicity that n_(o) (λ_(i))˜1. From Equation (40) the following conclusions can now be drawn:

(a) Under collision-dominated conditions in accordance with the invention the intensity requirement of I₁ is independent of the intensity of the beam of frequency ω₂.

It is to be noted here that the assumption that the main intensity change in E₁ (and E'₁) is due to absorption is valid only when 2|β₂ |ω₁ |E₂ |² /c<<γ(ω₁) [see Equations (20c) and (20d)]. This puts an upper limit on the intensities |E₂ |² and |E'₂ |² for which Equations (25a) and (25b) are valid. In the following discussion it is assumed that |E₂ |², |E'₂ |² are below the above noted limit.

(b) For large diffusion coefficient D, (i.e., when the refractive index modulation is dissipated mainly by diffusion), the intensity requirement of Equation (40) is independent of the lifetime τ₁₁ of level 1.

(c) For λ₂ >λ₁, the intensity requirement for total deflection is more moderate than for λ₂ <λ₁. In fact if λ₂ >λ₁, then selecting the angle α of the fields that generate the grating (see FIG. 2) such that

    α=λ.sub.1 /λ.sub.2

will reduce the right-hand side of inequality in Equation (34) to zero. The corresponding Bragg angle θ in this case is 90° and for this angle Equation (28) has to be used.

Consider now the power requirements as a function of α for a typical case, first when λ₂ <λ₁ and then when λ₂ >λ₁. For the case where λ₂ <λ₁, typical operating conditions may be assumed to be:

λ₁ =5000 A°

λ₂ =4000 A°

f₁₂ =0.1

υ₁₀ =2×10⁴ cm⁻¹

υ₂₁ =2.5×10⁴ cm⁻¹

Δ₀ =10 cm⁻¹

τ₁₁ =10 nsec

D=0.1 cm² /sec (1 Atm gas)

Using the above noted assumptions the intensity requirement as a function of the angle α is given immediately herein below in Table 1.

                  TABLE 1                                                          ______________________________________                                         α      2θ                                                                              I.sub.1 > (MW/cm.sup.2)                                    ______________________________________                                          6°   9.6°                                                                           0.15                                                       30°   47°                                                                            1.6                                                        60°   88°                                                                            6                                                          ______________________________________                                    

The deflection angle 2θ of the deflected beam appropriate for the specified value of α is given in the second column of Table 1 above. The increase in the intensity requirements with increasing α is mainly because of the dissipating effect of diffusion as discussed before.

For the case where λ₂ >λ₁, typical operating conditions may be assumed to be:

λ₁ =5000 A°

λ₂ =1μ

f₁₂ =0.1

υ₁₀ =2·10⁴ cm⁻¹

υ₂₁ =10⁴ cm⁻¹

Δ₀ =10 cm⁻¹

τ₁₁ =10 nsec

D=0.1 cm² /sec

Again, using its above noted assumptions the corresponding intensity requirement is given immediately hereinbelow in Table 2.

                  TABLE 2                                                          ______________________________________                                         α      2θ                                                                              I.sub.1 >(MW/cm.sup.2)                                     ______________________________________                                          6°    24°                                                                           0.15                                                       20°    86 °                                                                          0.6                                                        25°   116°                                                                           0.66                                                       28°   140°                                                                           0.5                                                        29°   152°                                                                           0.4                                                        30°   180°                                                                           --                                                         ______________________________________                                    

From the two examples set forth above it may now be seen that total deflection of a laser at ω₂ in an appropriate gaseous medium is possible using moderate intensities of the field at ω₁, the power requirement increasing with increasing deflection angle. However, this increase is more moderate if one selects for a given ω₂ an appropriate gaseous medium and ω₁ such that λ₁ <λ₂.

A real-time holographic curved reflector or a lens in accordance with the invention for a laser at frequency ω₂ can be generated by interfering two mutually coherent beams at frequency ω₁ having appropriate phase fronts in an appropriate nonlinear dispersive gaseous medium more fully discussed hereinafter.

Directing attention now to FIG. 4, a real-time holographic lens for frequency ω₂ can be produced for example by interfering in the medium two mutually coherent beams at frequency ω₁ where one [ε₁ (ω₁)] is a plane wave and the other [ε₁ '(ω₁)] is a spherical wave as shown in FIG. 4. This interference results in surfaces of constant intensity, which are paraboloids of revolution around the optical axis given by:

    R.sup.2 =2Zmλ.sub.1 +(mλ.sub.1).sup.2        (41)

where m is a positive integer (0,1,2 . . . ) and R is the radius vector in the xy plane. This intensity modulation at ω₁ produces in turn a real-time refractive index modulation for light at frequency ω₂. A beam ε₂ (ω₂) at frequency ω₂ impinging from the right will be condensed by the lens while a similar beam impinging from the left will diverge, provided that (a) the Bragg condition is satisfied for its entire cross section, and (b) the interaction is strong enough to divert the beam at ω₂ completely.

Since the real-time holographic lens is generated by interference of beams of wavelength λ₁, then according to the principle of holographic reconstruction, the Bragg condition is fully satisfied for a beam at wavelength λ₁. Clearly the Bragg condition will also be satisfied by λ₂ if

    λ.sub.1 n.sub.o (λ.sub.2)=λ.sub.2 n.sub.o (λ.sub.1)                                          (42)

The only way that this can be satisfied for λ₁ ≠λ₂ is if linear anomalous dispersion at λ₁ or λ₂ will be large enough to compensate for the difference in the wavelengths.

Referring to FIG. 1, for off-resonance conditions on 0→1 transition, i.e., |δ₀ |>>1/τ₁₀ : ##EQU34## and Δλ₁ is the difference between λ₁ and the wavelength for resonance transition on 0→1. R₀ is the classical radius of the electron (2.8×10⁻¹³ cm).

Similarly, ##EQU35## Inserting into Equation (42) and assuming that λ₂ is far from resonance for the 0→1 transition so that we can neglect B/Δλ₂ : ##EQU36## where Δλ=λ₁ -λ₂.

Taking as an example f₀₁ =0.5, υ₁₀ =1×10⁴ cm⁻¹, and Δλ₁ ˜1 Å:

    Δλ(Å)=7×10.sup.-16 ×N

For Δλ=700 Å one needs a density of 10¹⁸ atoms/cm³. It should be mentioned here that in principle it is possible to satisfy Equation (42), by using a mixture of two gases, where one gas is the active one for the nonlinear processes discussed above, and the other gas is a background gas providing the necessary linear dispersion to satisfy Equation (42).

When the radius of curvature of the constant refractive index lines in the yz plane is large compared to the beam diameter, one can approximate the lens to the planar grating earlier discussed herein in detail. In this case, provided that the Bragg condition is satisfied by use of linear dispersion in gas mixtures as discussed above, the intensity requirement for efficient operation of the lens is similar to that for a grating, as given by Equation (40).

Gaseous mediums suitable for purposes of the present invention include an atomic or molecular principal gas in combination with a buffer gas. Buffer gases may include inert gases such as, for example, helium, argon, xenon, krypton and the like. While, as noted above, the present invention contemplates the use of molecular gases, best results are more likely to be obtained with atomic gases such as, for example, alkali gases including sodium, rubidium, potassium and the like.

Broadly, a buffer gas or gases are suitable that spectrally broaden the revelent transitions of the principal gas and which does not interfere with previously discussed functions of the principal gas.

By way of example, in accordance with the invention a suitable gaseous medium may be maintained at a total pressure of about one tenth atmosphere to one atmosphere or more and comprise an atomic gas at about one thousandth atmosphere and a buffer gas or gases comprising the balance of the pressure and gaseous medium.

Specific pressures and mixture of gases will vary since for a given application gases and laser wavelengths are chosen that the laser beam wavelength or wavelengths at least substantially match revelent primary transitions, i.e., near resonance conditions as discussed in connection with FIG. 1 are obtained.

Typical apparatus forming the means to retain a suitable gaseous medium to define a grating, lens, prism or the like in accordance with the invention is shown in FIG. 5. Where an alkali gas such as, for example, sodium is used a gas cell 10 having oppositely disposed windows 11 and 12 transparent to the laser beams being used is disposed in a conventional oven 13 adapted to maintain the gaseous medium at a suitable temperature of, for example, about 300° C.

The oven 13 is also provided with windows 14 and 15 similar to and coaxial with windows 11 and 12 to permit the laser beams as previously described to enter and leave or be reflected from the gas cell 10 as the case may be. If the gas cell is heated directly the oven may be omitted.

The gas cell may be provided with a conventional cold finger 20 communicating with the interior of the gas cell and containing a sodium pellet 21 or the like. Coil 22 designates any suitable and conventional means for either adding or removing heat from the cold finger 20 as circumstances may require to maintain the temperature within cold finger 20 at a desired constant temperature several degrees less than the temperature within the main portion 23 of the gas cell 10. The interior of the gas cell is selectively coupled to a vacuum source to maintain the interior of the gas cell at the desired pressure in the event it is to be less than atmospheric and is also selectively coupled to a suitable source of buffer gas. The interior of the oven 13 may also be coupled to a vacuum source to prevent the existence of any undesirable gases between windows 14 and 11 and windows 12 and 15.

Whereas the gas cell 10 of FIG. 5 has been shown having curved top and bottom walls 25 and 26 the invention is not so limited and such walls may be flat or otherwise configured as may be required or desired. Regarding selection of the length of the reaction region, i.e., the distance between windows 11 and 12, such is discussed in connection with FIGS. 2 and 3.

Considering now the preceding discussion relating the holographic optics the interference of two coherent laser beams at wavelength λ₁, in accordance with the invention will generate a realtime refractive index volumetric grating in the gaseous medium for a laser beam at wavelength λ₂. If the phase fronts of the laser beams at wavelength λ₁ are flat a volumetric grating is generated consisting of planes of constant refractive index with sinusoidal modulation of the refractive index perpendicular to those planes as illustrated in FIG. 2.

A laser beam at wavelength λ₂ will consequently be coherently diffracted by the grating provided its angle of incidence to the planes of constant refractive index satisfies the Bragg condition. Such a grating can be generated as a transmission grating where the laser beam at wavelength λ₂ is deflected or reflected at up to 180°.

It has been previously pointed out that the response time of the gaseous medium is dependent on the lifetimes involved and gaseous diffusion, and can be made as short as a nanosecond. Thus the grating may also operate as a fast switch for high power lasers, the response time being determined by the longer of the gaseous response time and the rise time of the controlling laser at wavelength λ₁. A further advantage is that such a switch does not necessitate a good beam quality of the controlling laser at wavelength λ₁. For a simple switch, transmission grating or reflection grating, only a single controlling laser beam at wavelength λ₁ is needed.

If the aforementioned two interfering laser beams have phase fronts as shown and discussed in connection with FIG. 4, for example, realtime spherical or curved reflectors or lenses will be generated. Thus, if a paraxial laser beam at wavelength λ₂ impinges on the gaseous medium from the right, it will be coherently diffracted by the realtime curved grating and will be focused at the origin (0,0), provided it satisfies the Bragg condition for its entire cross section. Similarly, if this laser beam impinges on the gaseous medium from the left, it will be coherently diffracted and diverge with the origin (0,0) as a virtual focal point.

In addition to the above, where the laser beam at wavelength λ₂ is at a higher power level than the laser beams at wavelength λ₁, a realtime holographic amplifier is produced.

Since the nonlinear dispersion process is a near-resonance effect, it is wavelength sensitive. Accordingly, if the gaseous medium comprises a mixture of three gases, each one of them interacting nonlinearly with one set of two wavelengths λ₁ :λ₂, a realtime color holographic display may be generated by choosing the three sets of λ:λ₂ such that the three wavelengths λ₂ corresponding to the three gases comprising the gaseous medium are in the range of the three base colors necessary for a color display.

Attention is now directed to FIG. 6 which shows aberration corrector means in accordance with the present invention for correcting phase front aberrations in a laser beam caused by passing the laser beam through means such as, for example, a laser amplifier.

It is of course, desirable in transmitting or amplifying any signal whether electrical, optical or acoustical, that the signal be processed with as little distortion as possible. Consider now a laser beam I₂ which by way of example, is amplified in a laser amplifier 30 which produces aberrations on the wavefront of the amplified output beam.

In accordance with the invention amplifier 30 forms one leg of an interferometer such as, for example, a Mach-Zender interferometer. Laser beam I₂ at wavelength λ₂ is introduced into the amplifier 30. A diffraction limited laser beam I₁ at wavelength λ₁ is caused to impinge on a beam splitter 32 whereby one half of it is reflected to mirror 31 and reflected therefrom into amplifier 30 coaxial with laser beam I₂. The other half of laser beam I₁ passes through beam splitter 32 and is reflected by mirror 33 to member 34 where it is reflected to member 35. The part of laser beam I₁ that goes through amplifier 30 picks up the same aberrations imposed on laser beam I₂. This portion of laser beam I₁ passes through dichroic member 34 which provides maximum transmission of laser beam I₁ where it combines and interferes with the "clean" or undistorted portion of laser beam I₁. Upon interference of the aforementioned two portions of the laser beam I₁ at member 34 at the output of the interferometer, the local phase front aberrations are translated to intensity modulations across the cross section of the now recombined laser beam I₁. It is necessary, of course, that prior to use the interferometer be aligned to produce full phase front interference, i.e., there are no fringes when there is no aberration.

The now recombined laser beam I₁ now carries the phase distortion information of amplifier 30 as intensity modulation across the beam cross section and is amplified where necessary in amplifier 36 to the required amplitude. Amplifier 36, where present, should add as little aberration as possible to the recombined laser beam I₁. Since phase corrector 37 more fully described hereinafter will provide phase lags of a fraction of a wavelength over its entire length of interaction, the power requirements of laser beam I₁ will be quite moderate and amplifier 36 can indeed be made to provide very little if any distortion.

After passing through amplifier 36 recombined laser beam I₁ via mirror 38 is introduced into phase corrector 37. Similarly, after passing through amplifier 30 laser beam I₂ via mirrors 39 and 40 is also introduced in phase corrector 37. Laser beam I₁ and I₂ are introduced collinearly but in opposite directions into phase corrector 37 which comprises a gas cell containing a gaseous medium as shown and described in connection with FIGS. 1 through 5. Recombined laser beam I₁ exits through dichroic member 40 and the corrected laser beam I₂ exits through dichroic member 38 as shown in FIG. 6.

In phase corrector 37, as previously described, the intensity modulation across the cross section of laser beam I₁ introduced therein is transformed by nonlinear dispersion interaction into reflective index variations at wavelength λ₂ which causes, in turn, corresponding phase changes across the phase front of laser beam I₂ which is at wavelength λ₂.

Changing the path length of one leg of the interferometer changes the phase of the refractive index modulation across the width of the phase corrector. Thus, proper adjustment of this path length results in adjustment of the phase corrector 37 to compensate and correct for the phase front aberrations imposed on laser beam I₂ by amplifier 30.

The response time of phase corrector 37 is, as previously noted, determined by the lifetimes and diffusion involves in the gaseous medium and can be as short as one nanosecond. Thus, a main advantage of an aberration corrector in accordance with the invention as discussed above is that it permits correction of rapid phase front aberrations. 

We claim:
 1. In the method of laser beam wave front phase compensation the steps comprising:(a) providing a gaseous medium comprising a mixture of a first gas and a buffer gas, said first gas having a first resonance transition and a second higher transition where both said transitions are allowed electric dipole transitions, said buffer gas substantially collisionally broadening said first and second transitions of said first gas; (b) maintaining said gaseous medium at a pressure whereby spectral broadening of said first and second transitions is collision-dominated; (c) providing a first controlling laser beam having a first wavelength and having substantially no phase front aberrations, said first wavelength being at least near resonance with said first transition; (d) providing a second controlling laser beam at said first wavelength and having substantially no phase front aberrations; (e) providing a controlled laser beam having a second wavelength at least near resonance with said second transitions; (f) directing said controlled laser beam along a path resulting in first aberrations of its phase front; (g) directing said first controlling laser beam along said path resulting in second aberrations of its phase front at least substantially the same as said first phase front aberrations; (h) combining said first controlling laser beam with said second aberrations with said second controlling laser beam whereby they interfere one with another and causing said combined controlling laser beams to be incident on said gaseous medium and induce a predetermined change in the refractive index of said gaseous medium at a wavelength at least near said second wavelength; and (i) causing said controlled laser beam to be incident on said gaseous medium collinear with said combined controlling laser beams.
 2. The method defined in claim 1 and additionally including providing a primary controlling laser beam and deriving said first and second controlling laser beams from said primary controlling beam.
 3. The method as defined in claim 1 wherein said path is at least in part through an amplifier for amplifying said controlled laser beam.
 4. The method as defined in claim 1 wherein said path resulting in aberrations of said first controlling laser beam forms a first leg of an interferometer and the path of said second controlling laser beam forms a second leg of said interferometer and said first and second controlling laser beams are each derived from a diffraction limited primary controlling laser beam.
 5. The method as defined in claim 4 wherein said path is at least in part through an amplifier for amplifying said controlled laser beam.
 6. The method as defined in claim 1 wherein in said combined controlling laser beam the phase front aberrations of said first controlling laser beam are translated to intensity modulations across the front of said combined controlling laser beam.
 7. The method as defined in claim 6 wherein said gaseous medium the said intensity modulations of said combined laser beam are transformed by nonlinear dispersion interaction in said gaseous medium into refractive index variations at said second wavelength which cause corresponding phase changes across the phase front of said controlled laser beam.
 8. The method as defined in claim 7 wherein the path length of one of said legs of said interferometer is adjusted whereby the phase of the refractive index modulation in said gaseous medium compensates for the first aberrations imposed on said controlled laser beam.
 9. Apparatus for laser beam wave front phase compensation comprising:(a) housing means including at least one window transparent to laser beam radiation; (b) a gaseous medium comprising a mixture of a first gas and a buffer gas disposed within said housing means, said first gas having a first resonance transition and a second higher transition where both said transitions are allowed electric dipole transitions, said buffer gas substantially collisionally broadening said first and second transitions of said first gas; (c) first means for maintaining said gaseous medium at a pressure within said housing means whereby spectral broadening of said first and second transitions is collision-dominated; (d) second means for providing a first controlling laser beam having a first wavelength and having substantially no phase front aberrations, said first wavelength being at least near resonance with said first transition; (e) third means for providing a second controlling laser beam at said first wavelength and having substantially no phase front aberrations; (f) fourth means for providing a controlled laser beam having a second wavelength at least near resonance with said second transitions; (g) first directing means for receiving said controlled laser beams and directing it along a path resulting in first aberrations of its phase front; (h) second directing means for receiving said first controlling laser beam and directing it along said path resulting in second aberrations of its phase front at least substantially the same as said first phase front aberrations; (i) combining means for receiving said first controlling laser beam with said second aberrations and said second controlling laser beam and combining them whereby they interfere one with another; (j) third directing means for receiving said combined controlling laser beam and causing it to be incident on said gaseous medium and induce a predetermined change in the refractive index of said gaseous medium at a wavelength at least near said second wavelength; and (k) fourth directing means for receiving said controlled laser beam from said path and causing it to be incident on said gaseous medium collinear with said combined controlling laser beams.
 10. Apparatus as defined in claim 9 and additionally including:(a) means for providing a primary controlling laser beam and; (b) means for deriving said first and second controlling laser beams from said primary controlling laser beam.
 11. Apparatus as defined in claim 9 wherein said second directing means defines one leg of an interferometer and couples said first controlling laser beam to said combining means and additionally including third directing means defining a second leg of said interferometer for receiving said second controlling laser beam and coupling it to said combining means.
 12. Apparatus as defined in claim 11 wherein in said combining means the phase front aberrations of said first controlling laser beam are translated to intensity modulations across the front of said combined controlling laser beam.
 13. Apparatus as defined in claim 12 wherein said gaseous medium transforms the said intensity modulations of said combined controlling laser beam by nonlinear dispersion interaction into refractive index variations at said second wavelength which cause corresponding phase changes across the phase front of said controlled laser beam.
 14. Apparatus as defined in claim 13 and additionally including means for changing the path length of one of said legs of said interferometer whereby the refractive index modulation in said gaseous medium compensates for the first aberrations impared on said controlled laser beam. 